Digraphs+Theory,+Algorithms+and+Applications_Part23 - 8.1...

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Unformatted text preview: 8.1 Underlying Graphs of Various Classes of Digraphs 423 B i A i C i C j C p A p B p A j B j Figure 8.3 Implication classes for orientations of a graph G as a local tournament digraph.The sets C i ,C j ,C p denote distinct connected components of G . For each component a bipartition A r ,B r is shown. The edges shown inside C i form one implication class and the edges shown between C j and C p form another implication class. Theorem 8.1.13 (Huang) [436] Let G be a proper circular-arc graph which is reduced (that is, every edge is unbalanced), let ¯ G denote the complement graph of G and let C 1 ,...,C k denote the connected components of ¯ G . (a) If ¯ G is not bipartite, then k = 1 and (up to a full reversal) G has only one orientation as a locally tournament digraph, namely the round ori- entation. (b) If ¯ G is bipartite then every orientation of G as a locally tournament digraph can be obtained from the round locally tournament digraph ori- entation D of G by repeatedly applying one of the following operations: (I) reverse all arcs in D that go between two different C i ’s, (II) reverse all arcs in D that have both ends inside some C i . ut It is also possible to derive a similar result characterizing all possible orientations of G as a locally semicomplete digraph. We refer the reader to [436] for the details. As an example of the power of Huang’s result (Theorems 8.1.12 and 8.1.13) we state and prove the following corollary which was implicitly stated in [436] (see also Exercise 4.33). Corollary 8.1.14 If D is a locally tournament digraph such that UG ( D ) is not bipartite, then D = R [ S 1 ,...,S r ] , where R is a round locally tournament digraph on r vertices and each S i is a strong tournament. Proof: If UG ( D ) is reduced, then this follows immediately from Theorem 8.1.13, because according to Theorem 8.1.13, there is only one possible lo- cally tournament digraph orientation of UG ( D ). So suppose that UG ( D ) is not reduced. By Lemma 8.1.11, UG ( D ) = H [ K a 1 ,...,K a h ], h = | V ( H ) | , where H is a reduced proper circular arc graph, each K a i is a complete graph, 424 8. Orientations of Graphs and some a i ≥ 2. Because we can obtain an isomorphic copy of H as a sub- graph of UG ( D ) by choosing an arbitrary vertex from each K a i , we conclude, from Theorem 8.1.12, that in D all arcs between two distinct K a i , K a j have the same direction (note that H is non-bipartite). Thus D = R [ S 1 ,...,S r ], where (up to reversal of all arcs) R is the unique round locally tournament digraph orientation of H and each S i is the tournament D h V ( K a i ) i . Note that D h V ( K a i ) i may not be a strong tournament, but according to Corol- lary 4.11.7 we can find a round decomposition of D so that this is the case....
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Digraphs+Theory,+Algorithms+and+Applications_Part23 - 8.1...

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